I've learned how to find $f(g(x))$ when given the two $f(x)$ and $g(x)$ functions fairly easily, but I haven't found anywhere online showing how to do the opposite. For this question I'm working on I'm asked to find $f(x)$ and $g(x)$ if $\cos^2(x) = f(g(x))$.
Can anyone please help me figure out how to solve this? Also if someone could provide a website that helps explain this that would be greatly appreciated.
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In general, you'd never be able to tell. However, there are 'more correct' answers than others. For example, you have the example $$ f(g(x))=\cos^2(x) $$ Notice that this is the same as $$ f(g(x))=\left(\cos x\right)^2 $$ So it would 'make sense' to choose $g(x)=\cos x$ and choose $f(x)=x^2$. However, There are more possible choices. For instance, choosing $g(x)=\sqrt{\cos x}$ and $f(x)=x^4$ would have also worked.
Furthermore, take the example of $$ f(g(x))=x $$ was $g(x)=2x$ and $f(x)=\frac{1}{2}x$? Or perhaps, $g(x)=\sqrt{x}$ and $f(x)=x^2$. Notice that one can uniquely determine the functions but there are smart and easy choices. It's just about choosing them 'wisely'.
This isn't possible to do uniquely, since for example $$ f\left(x\right)=x $$ and $$ g\left(x\right)=\cos^{2}x $$ gives you the desired result. However, I think the answer they are looking for is $$ f\left(x\right)=x^{2} $$ and $$ g\left(x\right)=\cos x $$ so that $$ f\left(g\left(x\right)\right)=\left(\cos x\right)^{2}\equiv\cos^{2}x. $$ Sounds like you have yourself a bad teacher/textbook.